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Dynamics of the Rational Difference Equation |
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PP: 45-53 |
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Author(s) |
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M. A. El-Moneam,
E. M. E. Zayed,
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Abstract |
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| In this article, we study the periodicity, the boundedness and the global stability of the positive solutions of the following
nonlinear difference equation
xn+1 = Axn +Bxn−k +Cxn−l +Dxn−s +
bxn−k +hxn−l
dxn−k +exn−l
, n = 0,1,2, .....,
where the coefficients A,B,C,D,b,d, e,h ∈ (0,¥), while k, l and s are positive integers. The initial conditions
x−s ,..., x−l ,..., x−k, ..., x−1, x0 are arbitrary positive real numbers such that k < l < s . We will prove that the equilibrium points
of this equation are locally asymptotically stable, global attractor and hence they are global stability. This equation will have ( or have
not ) prime period two solution under suitable conditions on these coefficients. The solutions of this equation will be proved to be
bounded. Some numerical examples will be given to illustrate our results. |
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